Locally Sierpinski Julia Sets of Weierstrass Elliptic ℘ Functions

Hawkins, Jane; Look, Daniel M.

doi:10.1142/s0218127406015453

Cited by 13 publications

(14 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Such an interesting mixture of entire and polynomial dynamics has been seen in some families recently [34]. In addition, some of the features we observed earlier, such as Sierpiński curve Julia sets, do arise in other families, such as the meromorphic maps known as the Weierstrass elliptic p-functions [38].…”

Section: Singular Perturbations Of Complex Polynomials 423supporting

confidence: 56%

Singular perturbations of complex polynomials

Devaney

2013

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. In this paper we describe the dynamics of singularly perturbed complex polynomials. That is, we start with a complex polynomial whose dynamics are well understood. Then we perturb this map by adding a pole, i.e., by adding in a term of the form λ/(z − a) d where the parameter λ is complex. This changes the polynomial into a rational map of higher degree and, as we shall see, the dynamical behavior explodes.One aim of this paper is to give a survey of the many different topological structures that arise in the dynamical and parameter planes for these singularly perturbed maps. We shall show how Sierpiński curves arise in a myriad of different ways as the Julia sets for these singularly perturbed maps, and while these sets are always the same topologically, the dynamical behavior on them is often quite different. We shall also describe a number of interesting topological objects that arise in the parameter plane (the λ-plane) for these maps. These include Mandelpinski necklaces, Cantor webs, and Cantor sets of circles of Sierpiński curve Julia sets.

show abstract

Section: Singular Perturbations Of Complex Polynomials 423supporting

confidence: 56%

Singular perturbations of complex polynomials

Devaney

2013

Bull. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…The Julia set of the Weierstrass elliptic ℘-function on any triangular lattice is always connected L. KOSS [14]; this family of functions contains examples where the Fatou set is nonempty as well as examples where the Julia set is the entire sphere. Additional topological properties of Julia sets in this family were studied in [16]. In [11], Hawkins proved that the Weierstrass elliptic function on any rhombic square lattice always has Julia set equal to the entire sphere and hence is connected.…”

Section: Theorem 11 If the Finite Critical Point Of A Quadratic Polmentioning

confidence: 99%

“…Results on how the lattice shape influences the dynamics of the Weierstrass elliptic function can be found in [10,11,12,13,14,15,16].…”

Section: Proposition 24 ([6])mentioning

confidence: 99%

A fundamental dichotomy for Julia sets of a family of elliptic functions

Koss

2009

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We investigate topological properties of Julia sets of iterated elliptic functions of the form g = 1/℘, where ℘ is the Weierstrass elliptic function, on triangular lattices. These functions can be parametrized by C − {0}, and they all have a superattracting fixed point at zero and three other distinct critical values. We prove that the Julia set of g is either Cantor or connected, and we obtain examples of each type.

show abstract

“…We study parameter space for Weierstrass elliptic ℘ functions with square period lattices since this restriction gives a family of elliptic functions which can be parametrized by a single nonzero complex parameter. On the other hand this class already exhibits most of the richness of behavior typical of elliptic functions as shown for example in [Hawkins & Koss, 2002, 2004 and [Hawkins & Look, 2005].…”

Section: Introductionmentioning

confidence: 98%

Parameterized Dynamics for the Weierstrass Elliptic Function Over Square Period Lattices

Hawkins

McClure

2011

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

We iterate the Weierstrass elliptic ℘ function in order to understand the dependence of the dynamics on the underlying period lattice L. We focus on square lattices and use the holomorphic dependence on the classical invariants (g2, g3) = (g2, 0) to show that in parameter space (g2-space) one sees both quadratic-like attracting orbit behavior and prepole dynamics. In the case of prepole parameters all critical orbits terminate at poles and the Julia set of ℘L is the entire sphere. We show that both the Mandelbrot-like dynamics and the prepole parameters accumulate on prepole parameters of lower order providing results on the dynamics occurring in parameter space "between Mandelbrot sets".

show abstract

Locally Sierpinski Julia Sets of Weierstrass Elliptic ℘ Functions

Cited by 13 publications

References 24 publications

Singular perturbations of complex polynomials

Singular perturbations of complex polynomials

A fundamental dichotomy for Julia sets of a family of elliptic functions

Parameterized Dynamics for the Weierstrass Elliptic Function Over Square Period Lattices

Contact Info

Product

Resources

About